Presumably your contract includes option X7, concerning delay damages, and Y(UK)2. Clause X7.1 requires that the contractor pays delay damages in certain circumstances.
Turning to the payment provisions, your project manager is required to assess the amount due (see clause 50.1) and certify that amount (see clause 51.1). The amount due includes for a deduction of any amount to be paid by or retained from the contractor (see 3rd bullet of clause 50.2), and that would include any amount calculated for delay damages under X7.1.
Therefore, you are correct to say that your project manager deals with the deduction of delay damages as part of the assessment of the amount due. The project manager’s certificate is the notified sum. You as employer do not then have to issue a separate pay-less notice for delay damages because they are already included in the notified sum of the amount due. JCT wording is very different to the NEC and specifically requires the employer, not the contract administrator, to make the deduction for liquidated damages.
In answer to your final question, your project manager needs to follow the payment processes in clauses 50, 51 and Y(UK)2 scrupulously. It is important to remember that the obligation to continue to make assessments and certify amounts due at the end of each assessment interval continues during the period between completion and 4 weeks after the defects certificate is issued (see clause 50.1). This includes issuing zero-value certificates. If this is not within the correct timescales an application made by the contractor could, in certain circumstances, become the notified sum because of the wording of the Act.
If you are in any doubt you should take further legal advice. But you will only need to worry about that if your project manager does what they are supposed to within the timescales set out in the contract.